A parallelogram \(ABCD\) and a point \(E\) are given. Through the points \(A, B, C, D\), lines parallel to the straight lines \(EC, ED, EA,EB\), respectively, are drawn. Prove that they intersect at one point.
Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.
Prove that a convex \(n\)-gon is regular if and only if it is transformed into itself when it is rotated through an angle of \(360^{\circ}/n\) with respect to some point.
Prove that the midpoints of the sides of a regular polygon form a regular polygon.
Two perpendicular straight lines are drawn through the centre of the square. Prove that their intersection points with the sides of a square form a square.
Two circles \(c\) and \(d\) are tangent at point \(B\). Two straight lines intersecting the first circle at points \(D\) and \(E\) and the second circle at points \(G\) and \(F\) are drawn through the point \(B\). Prove that \(ED\) is parallel to \(FG\).
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).
What is the minimum number of points necessary to mark inside a convex \(n\)-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?
Several chords are drawn through a unit circle. Prove that if each diameter intersects with no more than \(k\) chords, then the total length of all the chords is less than \(\pi k\).
Several circles, whose total length of circumferences is 10, are placed inside a square of side 1. Prove that there will always be some straight line that crosses at least four of the circles.